Are you Saudi? Will you be in western attire? All passengers have been to the US before?
I suspect the major issue is how quickly immigrations and customs agent process your family.

at this point, just fly as ticketed and hope for the best.
i can’t imagine AC having an agent at Bradley recheck, but suspect the UA agent that will for sure be there could tag it.

i’ve gotten plenty of airlines to reconfirm on new flights over less than a 10 minute sched change.